Abstract: A connected graph G = (V,E) of order atleast two, with order p and size q is called vertex-graceful if there exists a bijection ? : V → { 1, 2, 3, ··· p } such that the induced function ?*: E → { 0, 1, 2, ··· q-1} defined by ?*(uv) = (?(u)+ ?(v)) (mod q) is a bijection. The bijection ? is called a vertex-graceful labeling of G. A subset S of the set of natural numbers N is called consecutive if S consists of consecutive integers. For any set X, a mapping ? : X → N $ is said to be consecutive if ?(X) is consecutive. A vertex-graceful labeling ? is said to be strong if the function ?1: E → N defined by ?1(e)= ?(u) + ?(v) for all edges e = uv in E forms a consecutive set. It is proved that one vertex union of odd number of copies of isomorphic caterpillars is vertex-graceful and any caterpillar is strong vertex-graceful. It is proved that a spider with even number of legs (paths) of equal length appended to each vertex of an odd cycle is vertex-graceful. It is also proved that the graph lA(mj,n) is vertex-graceful for both n and l odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi. Further, it is proved that the graph A(mj, n) is strong vertex-graceful for n odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi.
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